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(1)

Internat.

J. Math.

&

Math. Sci. Vol. i0

No.

2

(1987)

381-390

RAYLEIGH WAVE SCATTERING

AT

THE

FOOT OF A MOUNTAIN

381

P.S. DESHWAL

and

K.K. MANN

Department of Mathematics Maharshi Dayanand University

Rohtak-124001, INDIA (Received March 3, 1986)

ABSTRACT. A theoretical study of scattering of seismic waves at the foot of a mountain

is discussed here. A mountain of an arbitrary shape and of width a (0

x

a, z

0)

in the surface of an elastic solid medium (z

0)

is hit by a Rayleigh wave The method of solution is the technique of Wiener and Hopf. The reflected, transmitted and scattered waves are obtained by inversion of Fourier transforms. The scattered waves behave as decaying cylindrical waves at distant points and have a large amplitude near the foot of the mountain. The transmitted wave decreases exponentially as its distance from the other end of the mountain increases.

KEY WORDS AND PHRASES.

Rayleigh

surface

waves,

wave

motions

in

elastic

solids,

reflected

and transmitted waves.

IgBO

AMS

SUBJECT CLASSIFICATION CODES.

73D20,

?3DIO.

I. INTRODUCTION.

Earthquakes throw a grave challenge to human beings. Seismic waves cause the havoc because of their scattering due to the inhomogeneities and discontinuities in the sur-face of the earth. Rayleigh waves, appearing on the surface of the earth, are responsi-ble for destruction of the buildings and loss of human lives. Scattering of seismic waves due to irregularities in the surface leads to large amplification and variation in ground motion during earthquakes.

The paper presents a mechanism by which the energy of a body wave is lessened by partial conversion into reflected body and surface waves and by scattering at the foot of a mountain.

(2)

382

medium is a liquid halfspace which reduces the number of elastic parameters, the basic equations and the boundary conditions. Not much theoretical study is available on the problem of scattering of elastic waves due to an irregular boundary of finite dlmenslon Recently Momol (1980, 82) has studied the problem of scattering of Rayleigh waves by semicircular and rectangular discontinuities in the surface of a solid halfspace. The solutions are not exact because of the approximations used in the solution of these problems.

We propose to discuss here the problem of scattering of Raylelgh waves at the foot of a mountain with its base occupying the region 0 x

!

a, z 0 in the surface of a solid halfspace z 0 Since we are interested in scattering of waves at the foot of a mountain, its shape is immaterial and it is assumed to be rigid such that there is no displacement across the mountain. The method of solution is the Fourier transformation of the basic equations and determination of unknown functions by the technique of Wiener

and Hopf (Noble,

(1958)).

2. STATEMENT OF THE PROBLEM.

The problem is two-dimenslonal in zx-plane. The mountain of width a has its

base along 0 x a, z 0 in the surface of an elastic solid halfspace z 0 (figure

I).

The medium is homogeneous, isotropic and slightly dissipative. If the re-tarding force of the medium is proportional to the velocity, then the wave equation is

2

2

2

+

E

+

(2.1)

x

2 3z2 c2

t

2 c 2

t

where c is the velocity of propagation and e > 0 is the damping constant. The pote tial function harmonic in time is

(x,z,t)

(x,z)

exp(-imt)

(2.2)

The equation

(2.1)

is now

(V

2

+

()2)

(x,z)

0

(2.3)

where

/ (m2

+

le)/c

I

+

i2

is complex whose imaginary part is small and

posi-tive.

An

incident wave

iP0X

-B0z

#i(x,z)

D(2p

k

2)

e e

(2.4)

iP0x

-60z

i(x’z)

D(21P080)

e e

(2.5)

where

P0

is a root of the Raylelgh frequency equation

f(p)

(2p

2 k

2)

4p286

0,

8

/

(p2

()2),

6

/

(p2

k2)

(2.6)

80

8(p0),

60

6(p0),

strikes at the foot of the mountain from the region x < 0. The potential

(x,z)

satisfies the wave equation

(V

2

+

k

2)

(x,z)

0

(2.7)

Let the total potentials be

#t(x’z)

(x,z)

+

#i(x’z)

(2.8)

(3)

RAYLEIGH

WAVE

SCATTERING

AT THE

FOOT OF

A

MOUNTAIN 383 The potentials are connected with the displacements (u,w) by

u

t/x

+

t/z

w

t/z

t/z

(2.10)

We assume that, for given z, (x,z) and (x,z) have the behaviour

exp(-dlxl)

as

Ix]

,

d > 0 The Fourier transform

_(p,z)

f0_

(x,z)

eipx dx, p a

+

ia

0

(2.11)

is analytic in the region n

0 < d of the complex p-plane. The Fourier transform ipx

#(p,z)

I

(x,z)

e dx ipx

la

ipx

I

0 (x,z) e dx

+

(x,z)

e dx

+

I

(x,z)

eipx dx

0 a

#_(p,z) +

#a(P’z)

+

+a(P’Z)

(2.12)

and its derivatives w.r.t.z are analytic in the strip -d < s

0 < d of the complex plane. The transform of

(x,z)

has the same behaviour.

3. BOUNDARY CONDITIONS.

The boundary conditions of the problem are

(i) u

#t/x

+

t/z

0 z 0 0 x a,

(3.1)

(ii) w

t/Sz

t/x

0 z 0 0

!

x

!

a,

(3.2)

(iii)

Zxkz

x

k2@

0 z 0 x

!

0 x

Z

a,

(3.3)

(iv)

2x,x

+

z)

+

k2

0 z 0 x

!

0 x a,

(3.4)

The conditions

(3.1)

and

(3.2)

subject to (2.3) and

(2.7)

reduce to

iPoX

-B0z

-i

-D(2p

k

2)

e e z 0, 0 x a,

(3.5)

iP0x

-60z

-i

-D(ZiP0B0)

e e z 0, 0

!

x

!

a,

(3.6)

4. SOLUTION OF THE PROBLEM.

Fourier transforms of the wave equations

(2.3)

and

(2.7)

lead to

d2Idz

2

82

0 8

+/-/

(p2

(i2)

(4.1)

d2/dz

2

62

0 6

+/-/

(p2

k2

(4.2)

Since

(x,z)

and

@(x,z)

are bounded as z tends to infinity and their transforms are also bounded. The solutions of

(4.1)

and

(4.2)

are

(p,z)

A(p)

e

-8z

(4o3)

@(p,z) B(p) e (4.4)

The signs in radicals for

8

and 6 are such that their real parts are positive for all p. We use the notations

(p)

@(p) for (p,0)

(p,O)

etc. From (4.3) we obtain
(4)

384 P.S.

DESHWAL

and K.K.

MANN

This equation is decomposed

(p)

#(-k)

#’_(-k)

]+

+(P)

+

(p)

(-k)

+

(k)

_

(p)

-

(4.6)

p+/k p-/k

/-2

/2(p-)

The left hand member of

(4.6)

is analytic in a

0 > -d and the right hand member in

a

0 > d They represent an entire function. Further each member tends to zero as

Pl

By Liouville’s theorem, the entire function is identically zero. Equating each member to zero, it is found that

+(p)

-

+(p)/8

+

qF(p),

q

(k)

’_(-k)

(4.7)

#_(p)

-#’_(p)/8

qF(p),

-q

’_(k)

i(-k)

F(p)

/2(p-)

/-2(p+)

Similarly, we find from

(4.4)

that

+(p)

-$+(p)/6

+

qG(p)

q

$(k)

’_(-k)

(4.8)

(4.9)

(4.10)

@_(p)

=-’(p)16

qG(p) -q

@’_(k)

(-k)

(4.11)

G(p)

(4.12)

/2k(p-k)

J-2k(p+k)

Fourier transforms of

(3.5)

and

(3.6)

lead to

D(2p

k

2)

i(p+p0)a

dx

i(p

+

p0

(e

I)

(4.13)

i(p+P0)X

a

(p)

-m(2p

k

2)

fa

e 0

a

(p)

-D

(2iP0B0)

la

i

(P+P0)

x

-2P080D

i(p+P0)

a

e dx

(e

i)

(4.14)

0

P+

P0

And those of

(3.1)

and

(3.2)

result in

i

(p+p0)

a

a(p)

2mP0B060(e

l)/(p+p0)

(4.15)

i

(p+p0)

a

(p)

-imB0(2P

k2)(e

l)/(p+p0)

(4.16)

We now multiply

(3.3)

by exp(ipx) and integrate between x and x 0 to

find out

2

f0

3__(8_

ipx

k2

f_0

eipX

8x 3z

x

)e

dx dx 0

(4.17)

Since w

3#/3z

3/3x

is continuous, the first integral in

(4.17)

is integrated by parts to obtain

3@)

ipx

k2

8

3

-2ip

/_0

(38z

x

e dx

_(p)

-2(z

x)0

or

8i

3i)

-2ipI(P)

+

(2p2- k2)

-(P)

(3z

x

(5)

RAYLEIGH WAVE SCATTERING AT THE

FOOT OF

A

MOUNTAIN 385 In the same way,

(3.4)

leads to

(2p2-k2)_(p)

+

2ip

’_(p)

2ip[(p-po)(2p-k2)

+

2P08060

Solving

(4.8)

(4.111

(4.18) and

(4.19)

for

_(p)

and

@_(p),

we find

(4.19)

f(p)_(p)

2ip6[q(2p 2- k

2)

G(p)

2ipSF(p)]

+

2iD(2p2-k2)[(p-p0)(2p-k2)

+

2P0B00

+4iPBoD(k2-2pp

O)

(4.20)

f(p)_(p)

(2p2-k2)[-2ipBF(p)

+

2BoD(k2-2pp

01]

-2ipB[2ip6qG(p)

+

2iD[(p-p0)(2p-k2)

+

2P08060

]]

(4.21)

Integrating

(3.3)

and

(3.4)

between x a and x after multiplying it by

exp(ipx), it is obtained that

i(p+po)a

(2P

2 -k

2)

+a(p)

2ip

a(p)

=-2oD(k2-

2pp0)

e

(4.22)

(2p2

-k21

La

(p)

+

2ip

-a

(p)

-2iD[

(p-p0)

(2p-k

21

+

2P0060].

exp (i

(p+p0)

a)

(4.23)

Inversion of Fourier transforms will give various waves.

5.

VARIOUS

WAVES.

The factor

exp(-ipx)

exp(-iex)

exp(eoX

in the inverse transforms makes the inte-grals vanish at infinity in the upper part of the complex plane if x < 0 and in the lower part if x > 0.

For

waves in the region x < O,

we have

$(x

z)

+ieo

-ipx

+ie0

$_(p,z)

e dp (5.1)

where the line integral

(5.1)

is in the strip -d < < d The contour is in the o

upper half of the complex plane where

_(-p,z)

is analytic and hence o

f

_(-p,z)

e-ipx dp

(5.2)

0

o From

(5.1)

and (5.2), it is found that

=’+iaO

,(x,z)

+ia0

[0_(p,z)

0_(-p,z)]

e

ipXdp

(5.3)

To find the integrand, the wave equation

(2.3)

is integrated from x to x 0 after multiplying it by exp(ipx), to find out

d2_/dz

2

82_

ffi-($/x)

0

+

ip()0

(5.4)

Changing p to -p and subtracting the resulting equation from (5.41, we obtain

(d2/dz

2

821[_(p,z)

_(-p,z)]

-2ipD(2p-k

21 exp(-80z)

(5.51

whose complete solution is

-Sz

2ipD(2p

-k2)

e-80z

(p,z)

_(-p,z)

E(p)

e

8z

+

E

l(p)

e

+

(5.6)

p2_

pg

(6)

386 P.S. DESHWAL and K.K.

MANN

(p z)

-

(-p z) 2Esinh

Bz +

4ip

[(2p2_k2)6q

G(p)

f(p)

+

D(2p 2-

k2)(2p

k

2)

+

2DB0k26]e

-Bz

-B0z

-Bz)

2_

p)

+

2ipD(2p

k

2)(e

e

/(p

Differentiating (5.7) w.r.t, z and putting z 0, we find

’(p)

’(-p)

2BE

4ip8f(p)

[(2p2_ k2)6

q G(p)

+

2DB0k2

(5.7)

2ipD(B-B

0)(2p

k

2)

+

D(2p 2- k

2)(2p

k

2)]

+

(5.8)

p2_p

2E is obtained from here as the left hand member is known from

(4.8)

and

(4.20).

We find the integrand in

(5.3)

to be

2ipD(B-B0)

(2P

-k2)

sinh

Bz

#_(p,z)

_(-p,z)

p2_p)

B

2ipD

(2p-k

2

-80z

-Bz

+

(e e

+

4ip[q(2p 2-

k2)6G(p)

p2_p

-8z

+

D(2p2-k2)(2p-k2)

+

2Dk2B06]

e (5.9) The pole at p

P0

contributes

4P060

(2p-k

2)

G(p

0)

+

2DB0(2p+k2)]

e

-ip0x

0

z

l(X,Z)

[0)

[q

e (5.10)

This represents the reflected wave in the region x < 0. This does not depend upon the width a of the mountain. The reflected shear wave in the region is found to be

4P0B

0

-iP0x

-6 z

l(X’Z)

[0)

[(2p-k

2)

F(Po)

4iPoBoD(2p-k2B060

)]

e e 0

(5.11)

The integrand in

(5.9)

has branch points at p k and p k. The branch cuts are given by the conditions

Re(B)

0

Re(6).

As discussed by

Ewtng

and Press

(1957),

the parts of branch cuts are hyperbolic as shown in figure 2. For contribution along the branch cut we put p

+

iu, u being small as the main contribution is around the branch point. Along the cut,

Re(B)

0 and

Im(8)

changes signs along two sides of the cut. Since

B

is imaginary,

82

is negative. Therefore

82

=(

+ iu)2_

()2

2iU(l

+

i2)

U2

(5.12)

From here

B

+

+i

i

0

(5.13)

Integrating (5.3) along two sides of the branch cut, we find

k2x

i e

I’[

_(p,z) -_(-p,z)]B: i

#2

(x’z)

2 0

--_

e-UX

-[_(p

z)

(-p z)]

B

_i]

du

2ek2

x

f

[H (u)

sin z
(7)

RAYLEIGH WAVE SCATTERING AT THE

FOOT OF

A MOUNTAIN

387

u is small,

Hi(u)

and

H2(u

are expanded around u 0 and only

HI(0)

and

H2(0)

are retained. The following Laplace integrals

(Oberhettinger,

(1973))

are used

sin

a

e-pt dt

aCn

exp(-a2/4p)/2p

3/2

(5.15)

cos a

t/{--e

-pt dt

(p/2

a2/4) exp(-a2/4p)/p

5/2

(5.16)

The scattered waves for the region x <0 are obtained to be

2

(x,z)

(2k--2)

312

z

[(2(k2)2

+

k

2)

+

2(2)2(2)2

+

k2).

x

(x

+

oz

2)

5/2

x

[q

G()(2(k--2)2

+

k

2)

/((2

)2

+

k

2)

280Dk2(2)2

+

k

2)

iD(2p

k2)(z(2)2+

k

2)].

exp

[k--2

(x+z

2

/2x)

(5.17)

when x

>>z,

then

r

/(x

2

+

z

2)

x+z

2/2x

(5.18)

and the scattered wave is of the form

2

(x,z)

GO

exp

(2r)/r

(5.19)

This represents a cylindrical wave. On the free surface

(z--0),

the wave has the behaviour of

exp(k2x)/x

3/2

We now find the waves transmitted to the other side of the mountain. The poten-tial for the region x > a is given by

_+ia -ipx

(x,z)

f_o+iao

+a(P’Z)

e dp

(5.20)

o

The

countour is in the lower half of the complex plane in which

La(-p,z)

is ana-lytic and so

.+ie 2ipa

e-ipx

0

J_+ieo

+a

(-p’z)

e dp

o From

(5.20)

and

(5.21),

we obtain

(5.21)

.+ie

[+a(p

z

+a(-p,z)]e

dp

(5.22)

(x,z)

-

=+ieo

e2ipa-

-ipx

o

We take the Fourier transform of the wave equation

(2.3)

from x a to x to

find out

d2+a/dZ2

82+a

(a/ax)

a e ipa

ip()

a e

ipa

(5.23)

Changing p to -p and subtracting the resulting equation from

(5.23)

it is found that

(d2/dz

2

82)[+a

(P

z)e

-ipa

+a(-p,z)e

ipa]

iPoa

-80z

2ipD(2p2-k

2)

e e

(5.24)

(8)

388

8z

+a

(p’z)

e2ipa.+a(-p,z)

D

l(p)

e

2ipD(2p-k

2 e e

p2_

p02

Using

(4.22)

and

(4.23)

and the procedure of

(5.6),

we find

i

(P+P0)

a

-80z

2ipD

(2pg-kZ)e

e

2ipa

(-p,z)

2D sinh

Bz

+a(P’Z)

e

T+a

p2_p

where

+

g(p)

exp(-Bz)/f(p)

-8z

+

D 2

(p)e

i(p+p0)a

-80z

-(5.25)

(5.26)

g(p)

+2ip[-(2p2-k

2)

qG(p)

(l+e

2ipa)

2ipBF(p)

(e2ipa-l)

2ipa

+

2pB0D(2p-k2)(p

0

+

_p0

2ipa

+

2P0

B0D(2p2-k

2)( I

e

P+P0

P-P0

e2ip

a

2p(2p-k2)BD(pO

+

e2ipa

2(2p-kP0B00D(p--O__

php

(5.27)

D in

(5.26)

is eliminated by the procedure for E in

(5.7).

It is obtained that

La(p,

z)

e2ipa--+a

(-p,z)

sinh8

Bz

[8F(p)

(l_e2ipa)

e2ipa

+iD(mp-k2)

(B-B0)

(P0

+

"P-P0

-)

i

(p+p0)a

-80z

-2ipm(2pg-k

2)

e e

/

(pZ_pg)

+

g(p)exp(-Bz)/f (p)

(5.28)

The pole P

-P0

in the integrand in

(5.22)

contributes

iP0x

-80z

3(x,z)

-m(2pg-k

2)

e e

4P0[q0(2pg-k2)

G(-P0)Cos2P0a

iP0

(x-a)

-B0z

-2P0B00qF(-P0)

sin2P0a

e e

/f’

(-p0)

(5.29)

The first term cancels exactly the incident wave and the second term is the transitional wave in the region x > a.

6.

CONCLUSIONS.

The integrals along the branch cuts_at p

-,

k lead to the scattered waves as obtained in

(5.19).

They behave as

exp(-2r)/r

where r is the distance from the scatterer. These are cylindrical waves which die at distant points from the foot of the mountain. On the free surface

(z

0),

the scattered waves have the form

exp(-2x)/x3/2

which is large at the points near the scatterer. Thus the energy of the scattered waves
(9)

RAYLEIGH WAVE SCATTERING AT THE

FOOT OF

A MOUNTAIN

389

transmitted wave in

(5.29)

depends upon width a of the mountain. As the distance from the other end of the mountain increases, the transmitted wave decreases exponentially and

dies out at distant points. The reflected waves are given by (5.10) and (5.11) which do

not depend upon the width of the mountain. Numerical results for the amplitude of the scattered wave have been computed

or

Poisson’s solids for which k

3

k at a point

(r

I/2

km, z

0)

in the region < 0 of the free surface. The results are obtained for q 0 and 1.8932 k. There is a sharp increase in the amplitude (fig.

3)

when the wave number k increases through small values.

The results of this paper have their application to underground nuclear explosions carried out on either side of the mountains llke Himalayas. It helps calculating the amount of energy reflected and scattered at the foot of the mountain and the amount of energy which is transmitted to the other side of the mountain.

X

z

C

(10)

6O

5O

4O

3O

2O

I0

/

-I0

-II

.12

-13

-14

-15

K

2

REFERENCES

i.

DESHWAL,

P.

S. Diffraction of a Compressional Wave by a Rigid Barrier in a Liquid Half-space, Pur9

AppI.

Geophs.,

85(1971),

107-124.

2.

DESHWAL,

P.

S. Diffraction of Compressional Waves by a Strip in a Liquid

Half-space,

Jr.

M@.

th.

Phys._ Sci., 15(1981),

263-281.

3.

EWING, W.

M.

JARDETSKY, W. S.,

and

PRESS, F.

Elastic

Waves

in Layered Media, McGraw Hill Book

Co.,

1957.

4.

KAZI, M.

H. The Lovewave Scattering Matrix for a Continental Margin, Geophys.

J. R.

Ast.

Soc.,

52(1978),

25-44.

5.

MOMOI,

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